Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

DOI10.1016/j.jcp.2018.10.045zbMath1415.68175OpenAlexW2899283552WikidataQ128976826 ScholiaQ128976826MaRDI QIDQ2314336

Paris Perdikaris, Maziar Raissi, George Em. Karniadakis

Publication date: 22 July 2019

Published in: Journal of Computational Physics (Search for Journal in Brave)

Full work available at URL: https://www.osti.gov/biblio/1595805

zbMATH Keywords

nonlinear dynamicsRunge-Kutta methodsmachine learningpredictive modelingdata-driven scientific computing

Mathematics Subject Classification ID

Learning and adaptive systems in artificial intelligence (68T05) Reaction-diffusion equations (35K57) Inverse problems for PDEs (35R30) NLS equations (nonlinear Schrödinger equations) (35Q55)

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\textit{hp}-VPINNs: variational physics-informed neural networks with domain decomposition, Physics-informed machine learning models for predicting the progress of reactive-mixing, Iterative surrogate model optimization (ISMO): an active learning algorithm for PDE constrained optimization with deep neural networks, A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics, Deep learning of thermodynamics-aware reduced-order models from data, Deep-learning-based surrogate flow modeling and geological parameterization for data assimilation in 3D subsurface flow, Non-invasive inference of thrombus material properties with physics-informed neural networks, Embedding data analytics and CFD into the digital twin concept, Spectral methods for nonlinear functionals and functional differential equations, Hierarchical deep-learning neural networks: finite elements and beyond, Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks, Continuous-time system identification with neural networks: model structures and fitting criteria, Deep autoencoder based energy method for the bending, vibration, and buckling analysis of Kirchhoff plates with transfer learning, A deep learning improved numerical method for the simulation of rogue waves of nonlinear Schrödinger equation, A compressed lattice Boltzmann method based on ConvLSTM and resnet, Rank-adaptive tensor methods for high-dimensional nonlinear PDEs, Model reduction and neural networks for parametric PDEs, Sympnets: intrinsic structure-preserving symplectic networks for identifying Hamiltonian systems, Machine learning for flux regression in discrete fracture networks, Gaussian process regression constrained by boundary value problems, POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition, Multi-fidelity meta modeling using composite neural network with online adaptive basis technique, PINN deep learning method for the Chen-Lee-Liu equation: rogue wave on the periodic background, Classification with Runge-Kutta networks and feature space augmentation, Urban planning image feature enhancement and simulation based on partial differential equation method, Objective-sensitive principal component analysis for high-dimensional inverse problems, Constructive deep ReLU neural network approximation, Estimating the time-dependent contact rate of SIR and SEIR models in mathematical epidemiology using physics-informed neural networks, A hybrid partitioned deep learning methodology for moving interface and fluid-structure interaction, Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks, Multi-fidelity regression using artificial neural networks: efficient approximation of parameter-dependent output quantities, PhyCRNet: physics-informed convolutional-recurrent network for solving spatiotemporal PDEs, HiDeNN-TD: reduced-order hierarchical deep learning neural networks, Mosaic flows: a transferable deep learning framework for solving PDEs on unseen domains, A novel sequential method to train physics informed neural networks for Allen Cahn and Cahn Hilliard equations, Physics-informed graph neural Galerkin networks: a unified framework for solving PDE-governed forward and inverse problems, A representative volume element network (RVE-net) for accelerating RVE analysis, microscale material identification, and defect characterization, Interpretable machine learning: fundamental principles and 10 grand challenges, A physics-informed multi-fidelity approach for the estimation of differential equations parameters in low-data or large-noise regimes, At the crossroads of simulation and data analytics, Deep learning of conjugate mappings, Supervised learning from noisy observations: combining machine-learning techniques with data assimilation, Algorithms of data generation for deep learning and feedback design: a survey, Data-driven peakon and periodic peakon solutions and parameter discovery of some nonlinear dispersive equations via deep learning, Weak adversarial networks for high-dimensional partial differential equations, Stability analysis of hierarchical tensor methods for time-dependent PDEs, Learning finite difference methods for reaction-diffusion type equations with FCNN, Data-driven discoveries of Bäcklund transformations and soliton evolution equations via deep neural network learning schemes, Learning constitutive relations from indirect observations using deep neural networks, Estimating adsorption isotherm parameters in chromatography via a virtual injection promoting double feed-forward neural network, Overcoming the curse of dimensionality for some Hamilton-Jacobi partial differential equations via neural network architectures, CENN: conservative energy method based on neural networks with subdomains for solving variational problems involving heterogeneous and complex geometries, Monte Carlo fPINNs: deep learning method for forward and inverse problems involving high dimensional fractional partial differential equations, Towards out of distribution generalization for problems in mechanics, NH-PINN: neural homogenization-based physics-informed neural network for multiscale problems, Neural-network based collision operators for the Boltzmann equation, Learning phase field mean curvature flows with neural networks, INN: interfaced neural networks as an accessible meshless approach for solving interface PDE problems, A physically-informed deep-learning model using time-reversal for locating a source from sparse and highly noisy sensors data, Data-driven solutions and parameter discovery of the Sasa-Satsuma equation via the physics-informed neural networks method, Stabilized reduced-order models for unsteady incompressible flows in three-dimensional parametrized domains, Solving PDEs by variational physics-informed neural networks: an a posteriori error analysis, Integrability and exact solutions of the (2+1)-dimensional KdV equation with Bell polynomials approach, Neural network-based parameter estimation of stochastic differential equations driven by Lévy noise, Error-correcting neural networks for semi-Lagrangian advection in the level-set method, Neural network approach to data-driven estimation of chemotactic sensitivity in the Keller-Segel model, The distortion of the Peregrine soliton under the perturbation in initial condition, Error analysis for physics-informed neural networks (PINNs) approximating Kolmogorov PDEs, Efficient coupled deep neural networks for the time-dependent coupled Stokes-Darcy problems, Stochastic modeling of inhomogeneities in the aortic wall and uncertainty quantification using a Bayesian encoder-decoder surrogate, A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method, Emulation of cardiac mechanics using graph neural networks, CAS4DL: Christoffel adaptive sampling for function approximation via deep learning, Lookback option pricing under the double Heston model using a deep learning algorithm, Committor functions via tensor networks, Multifidelity data fusion in convolutional encoder/decoder networks, A non-gradient method for solving elliptic partial differential equations with deep neural networks, Physics-informed neural networks for shell structures, A Bayesian approach for data-driven dynamic equation discovery, Physics-informed neural networks for gravity field modeling of small bodies, Solving partial differential equation based on extreme learning machine, Deep learning characterization of brain tumours with diffusion weighted imaging, Schwarz waveform relaxation-learning for advection-diffusion-reaction equations, Optimal control of PDEs using physics-informed neural networks, Multi-Fidelity Machine Learning Applied to Steady Fluid Flows, A Model-Constrained Tangent Slope Learning Approach for Dynamical Systems, Reconstruction of three-dimensional turbulent flow structures using surface measurements for free-surface flows based on a convolutional neural network, Sparse Deep Neural Network for Nonlinear Partial Differential Equations, Stationary Density Estimation of Itô Diffusions Using Deep Learning, Reduced basis methods for time-dependent problems, Asymptotic-preserving schemes for multiscale physical problems, Convergence of Physics-Informed Neural Networks Applied to Linear Second-Order Elliptic Interface Problems, A deep neural network-based numerical method for solving contact problems, Discretization of parameter identification in PDEs using neural networks, SympOCnet: Solving Optimal Control Problems with Applications to High-Dimensional Multiagent Path Planning Problems, Feedforward Neural Networks and Compositional Functions with Applications to Dynamical Systems, A Deep Learning Modeling Framework to Capture Mixing Patterns in Reactive-Transport Systems, High Order Deep Neural Network for Solving High Frequency Partial Differential Equations, VPVnet: A Velocity-Pressure-Vorticity Neural Network Method for the Stokes’ Equations under Reduced Regularity, An Augmented Lagrangian Deep Learning Method for Variational Problems with Essential Boundary Conditions, Physically motivated structuring and optimization of neural networks for multi-physics modelling of solid oxide fuel cells, Blood and breath alcohol concentration from transdermal alcohol biosensor data: estimation and uncertainty quantification via forward and inverse filtering for a covariate-dependent, physics-informed, hidden Markov model*, A Deep Learning Method for Elliptic Hemivariational Inequalities, Modern Koopman Theory for Dynamical Systems, Convergence Rate Analysis for Deep Ritz Method, Deep Unfitted Nitsche Method for Elliptic Interface Problems, A Rate of Convergence of Physics Informed Neural Networks for the Linear Second Order Elliptic PDEs, Physics Informed Neural Networks (PINNs) For Approximating Nonlinear Dispersive PDEs, Imaging conductivity from current density magnitude using neural networks*, Approximating Optimal feedback Controllers of Finite Horizon Control Problems Using Hierarchical Tensor Formats, Neural Parametric Fokker--Planck Equation, Active Neuron Least Squares: A Training Method for Multivariate Rectified Neural Networks, slimTrain---A Stochastic Approximation Method for Training Separable Deep Neural Networks, The Discovery of Dynamics via Linear Multistep Methods and Deep Learning: Error Estimation, \(\text{PIN}^{\mathcal L}\)  : Preconditioned Inexact Newton with Learning Capability for Nonlinear System of Equations, Efficient Time-Stepping for Numerical Integration Using Reinforcement Learning, A deep learning method for solving high-order nonlinear soliton equations, A machine learning approach to calculating the non-equilibrium diffusion coefficients in the state-to-state solution of the Navier-Stokes equations, Time-dependent Dirac equation with physics-informed neural networks: computation and properties, Data-driven synchronization-avoiding algorithms in the explicit distributed structural analysis of soft tissue, Physics-informed machine learning for surrogate modeling of wind pressure and optimization of pressure sensor placement, A data-driven multi-flaw detection strategy based on deep learning and boundary element method, A physics-informed neural network technique based on a modified loss function for computational 2D and 3D solid mechanics, A neural network-based approach for bending analysis of strain gradient nanoplates, Deep learning neural networks for the third-order nonlinear Schrödinger equation: bright solitons, breathers, and rogue waves, A neural networks-based numerical method for the generalized Caputo-type fractional differential equations, A machine learning method for computing quasi-potential of stochastic dynamical systems, Solving second-order nonlinear evolution partial differential equations using deep learning, Learning-based local weighted least squares for algebraic multigrid method, Multifidelity deep operator networks for data-driven and physics-informed problems, A unified scalable framework for causal sweeping strategies for physics-informed neural networks (PINNs) and their temporal decompositions, JAX-DIPS: neural bootstrapping of finite discretization methods and application to elliptic problems with discontinuities, Physics-informed neural networks for 2nd order ODEs with sharp gradients, Numerical Solution of the Incompressible Navier-Stokes Equation by a Deep Branching Algorithm, Neural Networks with Local Converging Inputs (NNLCI) for Solving Conservation Laws, Part I: 1D Problems, Seq-SVF: an unsupervised data-driven method for automatically identifying hidden governing equations, Neural networks based on power method and inverse power method for solving linear eigenvalue problems, Simultaneous neural network approximation for smooth functions, Implicit integration of nonlinear evolution equations on tensor manifolds, VI-DGP: a variational inference method with deep generative prior for solving high-dimensional inverse problems, A singular Riemannian geometry approach to deep neural networks. II: Reconstruction of 1-D equivalence classes, Deep learning-accelerated computational framework based on physics informed neural network for the solution of linear elasticity, One-dimensional ice shelf hardness inversion: clustering behavior and collocation resampling in physics-informed neural networks, PhySR: physics-informed deep super-resolution for spatiotemporal data, Can cancer cells inform us about the tumor microenvironment?, MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEs, Frame Invariance and Scalability of Neural Operators for Partial Differential Equations, Solving Time Dependent Fokker-Planck Equations via Temporal Normalizing Flow, Microstructure-informed probability-driven point-particle model for hydrodynamic forces and torques in particle-laden flows, A Phase Shift Deep Neural Network for High Frequency Approximation and Wave Problems, Numerical solution of inverse problems by weak adversarial networks, Neural network representation of the probability density function of diffusion processes, Unsupervised deep learning for super-resolution reconstruction of turbulence, Convolutional Neural Networks in Phase Space and Inverse Problems, DeepXDE: A Deep Learning Library for Solving Differential Equations, Discovery of Dynamics Using Linear Multistep Methods, Unnamed Item, Deep neural networks for waves assisted by the Wiener–Hopf method, Learning on dynamic statistical manifolds, Locally adaptive activation functions with slope recovery for deep and physics-informed neural networks, SINDy-PI: a robust algorithm for parallel implicit sparse identification of nonlinear dynamics, Bayesian differential programming for robust systems identification under uncertainty, Finite Neuron Method and Convergence Analysis, Multi-Scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains, Extended Physics-Informed Neural Networks (XPINNs): A Generalized Space-Time Domain Decomposition Based Deep Learning Framework for Nonlinear Partial Differential Equations, On the Convergence of Physics Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs, Multi-Scale Deep Neural Network (MscaleDNN) Methods for Oscillatory Stokes Flows in Complex Domains, Learning to Discretize: Solving 1D Scalar Conservation Laws via Deep Reinforcement Learning, An Adaptive Surrogate Modeling Based on Deep Neural Networks for Large-Scale Bayesian Inverse Problems, Train Like a (Var)Pro: Efficient Training of Neural Networks with Variable Projection, DL-PDE: Deep-Learning Based Data-Driven Discovery of Partial Differential Equations from Discrete and Noisy Data, Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks, Deep Nitsche Method: Deep Ritz Method with Essential Boundary Conditions, The model reduction of the Vlasov–Poisson–Fokker–Planck system to the Poisson–Nernst–Planck system via the Deep Neural Network Approach, Physics-Informed Neural Networks with Hard Constraints for Inverse Design, Unnamed Item, Solving Fokker-Planck equation using deep learning, fPINNs: Fractional Physics-Informed Neural Networks, PDE-Aware Deep Learning for Inverse Problems in Cardiac Electrophysiology, A Fast Time-Stepping Strategy for Dynamical Systems Equipped with a Surrogate Model, A Local Deep Learning Method for Solving High Order Partial Differential Equations, Approximative Policy Iteration for Exit Time Feedback Control Problems Driven by Stochastic Differential Equations using Tensor Train Format, Spatial and spectral characteristics of information flux between turbulent boundary layers and porous media, Deep neural networks can stably solve high-dimensional, noisy, non-linear inverse problems, Solving multiscale elliptic problems by sparse radial basis function neural networks, Deep Neural Networks for Solving Large Linear Systems Arising from High-Dimensional Problems, Control variate method for deep BSDE solver using weak approximation, On the approximation of functions by tanh neural networks, On a neural network approach for solving potential control problem of the semiclassical Schrödinger equation, De Rham compatible deep neural network FEM, On the order of derivation in the training of physics-informed neural networks: case studies for non-uniform beam structures, Controlling mean exit time of stochastic dynamical systems based on quasipotential and machine learning, Dynamic analysis on optical pulses via modified PINNs: soliton solutions, rogue waves and parameter discovery of the CQ-NLSE, Numerical solution of nonlinear stochastic differential equations with fractional Brownian motion using fractional-order Genocchi deep neural networks, Randomized neural network with Petrov-Galerkin methods for solving linear and nonlinear partial differential equations, Application of the dynamical system method and the deep learning method to solve the new (3+1)-dimensional fractional modified Benjamin-Bona-Mahony equation, Physics-informed neural networks for the Reynolds-averaged Navier-Stokes modeling of Rayleigh-Taylor turbulent mixing, Accelerated offline setup of homogenized microscopic model for multi‐scale analyses using neural network with knowledge transfer, Physics-informed neural networks with parameter asymptotic strategy for learning singularly perturbed convection-dominated problem, Probabilistic partition of unity networks for high‐dimensional regression problems, Laplace based Bayesian inference for ordinary differential equation models using regularized artificial neural networks, An introduction to the mathematics of deep learning, A convolutional dispersion relation preserving scheme for the acoustic wave equation, Three ways to solve partial differential equations with neural networks — A review, Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolution, Combining machine learning and domain decomposition methods for the solution of partial differential equations—A review, A perspective on machine learning methods in turbulence modeling, Physics-Informed Neural Networks for Solving Dynamic Two-Phase Interface Problems, VC-PINN: variable coefficient physics-informed neural network for forward and inverse problems of PDEs with variable coefficient, A neural network‐enhanced reproducing kernel particle method for modeling strain localization, Deep learning methods for partial differential equations and related parameter identification problems, Enhanced physics‐informed neural networks for hyperelasticity, Deep learning phase‐field model for brittle fractures, Deep learning‐based reduced order models for the real‐time simulation of the nonlinear dynamics of microstructures, CoolPINNs: a physics-informed neural network modeling of active cooling in vascular systems, A priori generalization error analysis of two-layer neural networks for solving high dimensional Schrödinger eigenvalue problems, A framework for machine learning of model error in dynamical systems, A physics-constrained deep residual network for solving the sine-Gordon equation, A dynamical neural network approach for solving stochastic two-player zero-sum games, Multibody dynamics and control using machine learning, Discrete-time nonlinear feedback linearization via physics-informed machine learning, Enforcing continuous symmetries in physics-informed neural network for solving forward and inverse problems of partial differential equations, On the recovery of internal source for an elliptic system by neural network approximation, Neural network representation of time integrators, A stepwise physics‐informed neural network for solving large deformation problems of hypoelastic materials, Multiscale modeling of linear elastic heterogeneous structures via localized model order reduction, A comparative study on different neural network architectures to model inelasticity, Phase-field DeepONet: physics-informed deep operator neural network for fast simulations of pattern formation governed by gradient flows of free-energy functionals, Synergistic integration of deep neural networks and finite element method with applications of nonlinear large deformation biomechanics, Solving nonconvex energy minimization problems in martensitic phase transitions with a mesh-free deep learning approach, Physical restriction neural networks with restarting strategy for solving mathematical model of thermal heat equation for early diagnose breast cancer, Discovering interpretable Lagrangian of dynamical systems from data, A data-driven Kaczmarz iterative regularization method with non-smooth constraints for ill-posed problems, Physics-agnostic and physics-infused machine learning for thin films flows: modelling, and predictions from small data, Determining the viscosity of the Navier–Stokes equations from observations of finitely many modes, Neural networks for first order HJB equations and application to front propagation with obstacle terms, Double and triple-pole solutions for the third-order flow equation of the Kaup-Newell system with zero/nonzero boundary conditions, Adaptive deep density approximation for fractional Fokker-Planck equations, The deep minimizing movement scheme, Physics-informed neural networks for approximating dynamic (hyperbolic) PDEs of second order in time: error analysis and algorithms, SONets: sub-operator learning enhanced neural networks for solving parametric partial differential equations, Model discovery of compartmental models with graph-supported neural networks, The Random Feature Method for Time-Dependent Problems, Development of data‐driven exponential integrators with application to modeling of delay photocurrents, Bayesian calibration for large‐scale fluid structure interaction problems under embedded/immersed boundary framework, Unsteady reduced order model with neural networks and flight-physics-based regularization for aerodynamic applications, A deep learning method for solving third-order nonlinear evolution equations, CD-ROM: complemented deep -- reduced order model, Automated model discovery for skin: discovering the best model, data, and experiment, BINN: a deep learning approach for computational mechanics problems based on boundary integral equations, Deep-OSG: deep learning of operators in semigroup, Adaptive weighting of Bayesian physics informed neural networks for multitask and multiscale forward and inverse problems, Dosnet as a non-black-box PDE solver: when deep learning meets operator splitting, DeepStSNet: reconstructing the quantum state-resolved thermochemical nonequilibrium flowfield using deep neural operator learning with scarce data, Computing non-equilibrium trajectories by a deep learning approach, Surrogate modeling of time-domain electromagnetic wave propagation via dynamic mode decomposition and radial basis function, A cusp-capturing PINN for elliptic interface problems, A dimension-augmented physics-informed neural network (DaPINN) with high level accuracy and efficiency, Fully probabilistic deep models for forward and inverse problems in parametric PDEs, Solving Elliptic Problems with Singular Sources Using Singularity Splitting Deep Ritz Method, Artificial neural network solver for time-dependent Fokker-Planck equations, Model-driven identification framework for optimal constitutive modeling from kinematics and rheological arrangement, Geometric learning for computational mechanics. III: Physics-constrained response surface of geometrically nonlinear shells, Deep Ritz method with adaptive quadrature for linear elasticity, A framework based on symbolic regression coupled with eXtended physics-informed neural networks for gray-box learning of equations of motion from data, On the use of neural networks for full waveform inversion, Physics-informed radial basis network (PIRBN): a local approximating neural network for solving nonlinear partial differential equations, Complex dynamics on the one-dimensional quantum droplets via time piecewise PINNs, Deep learning data-driven multi-soliton dynamics and parameters discovery for the fifth-order Kaup-Kuperschmidt equation, JAX-fluids: a fully-differentiable high-order computational fluid dynamics solver for compressible two-phase flows, Physics-informed deep learning for simultaneous surrogate modeling and PDE-constrained optimization of an airfoil geometry, FluxNet: a physics-informed learning-based Riemann solver for transcritical flows with non-ideal thermodynamics, Reliable extrapolation of deep neural operators informed by physics or sparse observations, A New Certified Hierarchical and Adaptive RB-ML-ROM Surrogate Model for Parametrized PDEs, Deep learning for thermal plasma simulation: solving 1-D arc model as an example, Viscous damping of steady-state resonant sloshing in a clean rectangular tank, Continuous limits of residual neural networks in case of large input data, Solving multi-material problems in solid mechanics using physics-informed neural networks based on domain decomposition technology, A fast inertial self-adaptive projection based algorithm for solving large-scale nonlinear monotone equations, Airfoil-based convolutional autoencoder and long short-term memory neural network for predicting coherent structures evolution around an airfoil, Exponential ReLU neural network approximation rates for point and edge singularities, Accuracy and architecture studies of residual neural network method for ordinary differential equations, \(\mathrm{FE^{ANN}}\): an efficient data-driven multiscale approach based on physics-constrained neural networks and automated data mining, Principled interpolation of Green's functions learned from data, Physically informed deep homogenization neural network for unidirectional multiphase/multi-inclusion thermoconductive composites, DeepBHCP: deep neural network algorithm for solving backward heat conduction problems, Detecting stochastic governing laws with observation on stationary distributions, Soliton and breather solutions of the higher-order modified Korteweg-de Vries equation with constants background, FDM-PINN: Physics-informed neural network based on fictitious domain method, A decoupled physics-informed neural network for recovering a space-dependent force function in the wave equation from integral overdetermination data, Radial basis function neural network (RBFNN) approximation of Cauchy inverse problems of the Laplace equation, Data-driven vortex solitons and parameter discovery of 2D generalized nonlinear Schrödinger equations with a \(\mathcal{PT}\)-symmetric optical lattice, The robust physics-informed neural networks for a typical fourth-order phase field model, Learning high frequency data via the coupled frequency predictor-corrector triangular DNN, Physics-informed deep learning for melting heat transfer analysis with model-based transfer learning, An extended physics informed neural network for preliminary analysis of parametric optimal control problems, Predicting rare events using neural networks and short-trajectory data, A study on data-driven identification and representation of nonlinear dynamical systems with a physics-integrated deep learning approach: Koopman operators and nonlinear normal modes, Data-driven passivity-based control of underactuated mechanical systems via interconnection and damping assignment, Pre-training strategy for solving evolution equations based on physics-informed neural networks, A method for computing inverse parametric PDE problems with random-weight neural networks, Physics-informed neural networks with adaptive localized artificial viscosity, Combining direct and indirect sparse data for learning generalizable turbulence models, Predicting continuum breakdown with deep neural networks, Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations, Multigrid-Augmented Deep Learning Preconditioners for the Helmholtz Equation, MGIC: Multigrid-in-Channels Neural Network Architectures, Friedrichs Learning: Weak Solutions of Partial Differential Equations via Deep Learning, Machine Learning Moment Closure Models for the Radiative Transfer Equation II: Enforcing Global Hyperbolicity in Gradient-Based Closures, A structure-preserving neural differential operator with embedded Hamiltonian constraints for modeling structural dynamics, Error estimates and physics informed augmentation of neural networks for thermally coupled incompressible Navier Stokes equations, Deep learning discrete calculus (DLDC): a family of discrete numerical methods by universal approximation for STEM education to frontier research, A Chebyshev Polynomial Neural Network Solver for Boundary Value Problems of Elliptic Equations, A Hybrid Method for Three-Dimensional Semi-Linear Elliptic Equations, Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation data, Solving an inverse source problem by deep neural network method with convergence and error analysis, An Efficient Neural-Network and Finite-Difference Hybrid Method for Elliptic Interface Problems with Applications, Investigating and Mitigating Failure Modes in Physics-Informed Neural Networks (PINNs), Deterministic neural networks optimization from a continuous and energy point of view, Accelerating explicit time-stepping with spatially variable time steps through machine learning, Stability estimate for a nonlinear coupled heat transfer model, Higher-order error estimates for physics-informed neural networks approximating the primitive equations, \textit{FastSVD-ML-ROM}: a reduced-order modeling framework based on machine learning for real-time applications, Error convergence and engineering-guided hyperparameter search of PINNs: towards optimized I-FENN performance, A multifidelity deep operator network approach to closure for multiscale systems, GRIDS-Net: inverse shape design and identification of scatterers via geometric regularization and physics-embedded deep learning, Efficient Natural Gradient Descent Methods for Large-Scale PDE-Based Optimization Problems, Port-Hamiltonian Dynamic Mode Decomposition, Parsimony as the ultimate regularizer for physics-informed machine learning, Neural Galerkin schemes with active learning for high-dimensional evolution equations, A deep learning method for the dynamics of classic and conservative Allen-Cahn equations based on fully-discrete operators, NAS-PINN: neural architecture search-guided physics-informed neural network for solving PDEs, A mathematical perspective of machine learning, An overlapping domain decomposition method for the solution of parametric elliptic problems via proper generalized decomposition, A shallow physics-informed neural network for solving partial differential equations on static and evolving surfaces, Conditional Karhunen-Loève regression model with basis adaptation for high-dimensional problems: uncertainty quantification and inverse modeling, Branched latent neural maps, Finite element interpolated neural networks for solving forward and inverse problems, A Bayesian defect-based physics-guided neural network model for probabilistic fatigue endurance limit evaluation, Capturing the diffusive behavior of the multiscale linear transport equations by asymptotic-preserving convolutional deeponets, Addressing discontinuous root-finding for subsequent differentiability in machine learning, inverse problems, and control, Pseudo-Hamiltonian neural networks for learning partial differential equations, HomPINNs: homotopy physics-informed neural networks for solving the inverse problems of nonlinear differential equations with multiple solutions, Variable linear transformation improved physics-informed neural networks to solve thin-layer flow problems, Existence and uniqueness of solution for a singular elliptic differential equation, Physics informed WNO, Extended tensor decomposition model reduction methods: training, prediction, and design under uncertainty, Adaptive task decomposition physics-informed neural networks, Solving seepage equation using physics-informed residual network without labeled data, An extreme learning machine-based method for computational PDEs in higher dimensions, Pseudo-Hamiltonian system identification, Neural network-based multiscale modeling of finite strain magneto-elasticity with relaxed convexity criteria, Physics-guided training of GAN to improve accuracy in airfoil design synthesis, Respecting causality for training physics-informed neural networks, A novel physics-informed deep learning strategy with local time-updating discrete scheme for multi-dimensional forward and inverse consolidation problems, Adversarial deep energy method for solving saddle point problems involving dielectric elastomers, Higher-order neurodynamical equation for simplex prediction, Limitations of neural network training due to numerical instability of backpropagation, Transferable neural networks for partial differential equations, Phase-amplitude coordinate-based neural networks for inferring oscillatory dynamics, Weak adversarial networks for solving forward and inverse problems involving 2D incompressible Navier-Stokes equations, Incremental neural controlled differential equations for modeling of path-dependent material behavior, Physics-based self-learning spiking neural network enhanced time-integration scheme for computing viscoplastic structural finite element response, Efficient physics-informed neural networks using hash encoding, Loss-attentional physics-informed neural networks, A conservative hybrid deep learning method for Maxwell-Ampère-Nernst-Planck equations, Koopman operator learning using invertible neural networks, Deep learning-based schemes for singularly perturbed convection-diffusion problems, Forward to the special topic on ``Solving differential equations with deep learning, NSNO: Neumann series neural operator for solving Helmholtz equations in inhomogeneous medium, Number of solitons emerged in the initial profile of shallow water using convolutional neural networks, A new method for solving nonlinear partial differential equations based on liquid time-constant networks, Pre-training physics-informed neural network with mixed sampling and its application in high-dimensional systems, Parallel physics-informed neural networks method with regularization strategies for the forward-inverse problems of the variable coefficient modified KdV equation, Physics-informed neural networks with two weighted loss function methods for interactions of two-dimensional oceanic internal solitary waves, A Review of Data‐Driven Discovery for Dynamic Systems, Random vibration of hysteretic systems under Poisson white noise excitations, Variational inference in neural functional prior using normalizing flows: application to differential equation and operator learning problems, Deep convolutional Ritz method: parametric PDE surrogates without labeled data, Gaussian process hydrodynamics, A dive into spectral inference networks: improved algorithms for self-supervised learning of continuous spectral representations, Forward sensitivity analysis and mode dependent control for closure modeling of Galerkin systems, SeismicNET: physics-informed neural networks for seismic wave modeling in semi-infinite domain, Model order reduction for parameterized electromagnetic problems using matrix decomposition and deep neural networks, Assessing the finite-time stability of nonlinear systems by means of physics-informed neural networks, Physics-informed neural networks for analysis of 2D thin-walled structures, A direct sampling-based deep learning approach for inverse medium scattering problems, DPK: Deep Neural Network Approximation of the First Piola-Kirchhoff Stress, A Rate of Convergence of Weak Adversarial Neural Networks for the Second Order Parabolic PDEs, Learning Specialized Activation Functions for Physics-Informed Neural Networks, Neural Networks with Local Converging Inputs (NNLCI) for Solving Conservation Laws, Part II: 2D Problems, A Variational Neural Network Approach for Glacier Modelling with Nonlinear Rheology, Deep neural networks learning forward and inverse problems of two-dimensional nonlinear wave equations with rational solitons, \(r\)-adaptive deep learning method for solving partial differential equations, Variable separated physics-informed neural networks based on adaptive weighted loss functions for blood flow model, Agglomeration of polygonal grids using graph neural networks with applications to multigrid solvers, Sobolev regularity of Gaussian random fields, Micromechanics-informed parametric deep material network for physics behavior prediction of heterogeneous materials with a varying morphology, Learning the nonlinear flux function of a hidden scalar conservation law from data, Error analysis of deep Ritz methods for elliptic equations, On the Poisson structure and action-angle variables for the Fokas-Lenells equation, Investigation of Low and High-Speed Fluid Dynamics Problems Using Physics-Informed Neural Network, Biomimetic IGA neuron growth modeling with neurite morphometric features and CNN-based prediction, Label-free learning of elliptic partial differential equation solvers with generalizability across boundary value problems, Quadrature rule based discovery of dynamics by data-driven denoising, Residual-based error correction for neural operator accelerated Infinite-dimensional Bayesian inverse problems, Direct Poisson neural networks: learning non-symplectic mechanical systems, Multi‐fidelity data fusion through parameter space reduction with applications to automotive engineering, Quantum Mechanics for Closure of Dynamical Systems, Convergence Analysis of a Quasi-Monte CarloBased Deep Learning Algorithm for Solving Partial Differential Equations, MC-Nonlocal-PINNs: Handling Nonlocal Operators in PINNs Via Monte Carlo Sampling, HRW: Hybrid Residual and Weak Form Loss for Solving Elliptic Interface Problems with Neural Network, PI-VEGAN: Physics Informed Variational Embedding Generative Adversarial Networks for Stochastic Differential Equations, Improved Analysis of PINNs: Alleviate the CoD for Compositional Solutions, A Discontinuity and Cusp Capturing PINN for Stokes Interface Problems with Discontinuous Viscosity and Singular Forces, Stochastic dynamics and data science, Multi-fidelity physics constrained neural networks for dynamical systems, Solving the Boltzmann Equation with a Neural Sparse Representation, NeuralUQ: A Comprehensive Library for Uncertainty Quantification in Neural Differential Equations and Operators, TNet: A Model-Constrained Tikhonov Network Approach for Inverse Problems, Bayesian Deep Learning Framework for Uncertainty Quantification in Stochastic Partial Differential Equations, Connections between numerical algorithms for PDEs and neural networks, Identification of the flux function of nonlinear conservation laws with variable parameters, Less is more: a new machine-learning methodology for spatiotemporal systems, Asymptotic-Preserving Neural Networks for multiscale hyperbolic models of epidemic spread, Boundary-safe PINNs extension: application to non-linear parabolic PDEs in counterparty credit risk, Double diffusion maps and their latent harmonics for scientific computations in latent space, Physics-informed variational inference for uncertainty quantification of stochastic differential equations, Numerical methods for backward stochastic differential equations: a survey, A deep neural network-based method for solving obstacle problems, An artificial neural network approach to bifurcating phenomena in computational fluid dynamics, Meshless methods for American option pricing through physics-informed neural networks, A connection element method: both a new computational method and a physical data-driven framework -- take subsurface two-phase flow as an example, DNN-HDG: a deep learning hybridized discontinuous Galerkin method for solving some elliptic problems, Asymptotic-preserving neural networks for multiscale time-dependent linear transport equations, Towards a machine learning pipeline in reduced order modelling for inverse problems: neural networks for boundary parametrization, dimensionality reduction and solution manifold approximation, Numerical computation of partial differential equations by hidden-layer concatenated extreme learning machine, ReLU neural network Galerkin BEM, \(\mathrm{U}^p\)-net: a generic deep learning-based time stepper for parameterized spatio-temporal dynamics, Bi-fidelity modeling of uncertain and partially unknown systems using DeepONets, HiDeNN-FEM: a seamless machine learning approach to nonlinear finite element analysis, An introduction to kernel and operator learning methods for homogenization by self-consistent clustering analysis, Embedding physical knowledge in deep neural networks for predicting the phonon dispersion curves of cellular metamaterials, Deep learning soliton dynamics and complex potentials recognition for 1D and 2D \(\mathcal{PT}\)-symmetric saturable nonlinear Schrödinger equations, Dissipative soliton dynamics of the Landau-Lifshitz-Gilbert equation, Approximation of compositional functions with ReLU neural networks, Discovery of PDEs driven by data with sharp gradient or discontinuity, Posteriori error neural network: a recovery type posteriori error estimator based on neural network for diffusion problems, PINN training using biobjective optimization: the trade-off between data loss and residual loss, Subspace decomposition based DNN algorithm for elliptic type multi-scale PDEs, Elasticity-mechanics-informed generative adversarial networks for predicting the thermal strain of thermal barrier coatings penetrated by CaO-MgO-\(\mathrm{Al_2O}_3\)-\(\mathrm{SiO}_2\), Convolution hierarchical deep-learning neural network (C-HiDeNN) with graphics processing unit (GPU) acceleration, Semi-supervised invertible neural operators for Bayesian inverse problems, On the geometry transferability of the hybrid iterative numerical solver for differential equations, Hybrid thermal modeling of additive manufacturing processes using physics-informed neural networks for temperature prediction and parameter identification, Physics-informed deep learning for three-dimensional transient heat transfer analysis of functionally graded materials, SPADE4: sparsity and delay embedding based forecasting of epidemics, Improved training of physics-informed neural networks for parabolic differential equations with sharply perturbed initial conditions, Automatic boundary fitting framework of boundary dependent physics-informed neural network solving partial differential equation with complex boundary conditions, PINN-FORM: a new physics-informed neural network for reliability analysis with partial differential equation, A symmetry group based supervised learning method for solving partial differential equations, Exact Dirichlet boundary physics-informed neural network EPINN for solid mechanics, Finite basis physics-informed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations, Distributed PINN for Linear Elasticity — A Unified Approach for Smooth, Singular, Compressible and Incompressible Media, Deep Weak Approximation of SDEs: A Spatial Approximation Scheme for Solving Kolmogorov Equations, Adaptive Learning Rate Residual Network Based on Physics-Informed for Solving Partial Differential Equations, Adaptive transfer learning for PINN, Physical information neural networks for 2D and 3D nonlinear Biot model and simulation on the pressure of brain, RelaxNet: a structure-preserving neural network to approximate the Boltzmann collision operator, Inverse modeling of nonisothermal multiphase poromechanics using physics-informed neural networks, A learned conservative semi-Lagrangian finite volume scheme for transport simulations, Sparse Gaussian processes for solving nonlinear PDEs, A Scalable Deep Learning Approach for Solving High-Dimensional Dynamic Optimal Transport, Failure-Informed Adaptive Sampling for PINNs, Least-squares neural network (LSNN) method for scalar nonlinear hyperbolic conservation laws: discrete divergence operator, Solving groundwater flow equation using physics-informed neural networks, Mobility Estimation for Langevin Dynamics Using Control Variates, A Neural Network Approach for Homogenization of Multiscale Problems, Learning Lagrangian fluid mechanics with E(3)-equivariant graph neural networks, On the spectral bias of coupled frequency predictor-corrector triangular DNN: the convergence analysis, Hard enforcement of physics-informed neural network solutions of acoustic wave propagation, Discovering efficient periodic behaviors in mechanical systems via neural approximators, Deep-learning-based upscaling method for geologic models via theory-guided convolutional neural network, Deep learning discovery of macroscopic governing equations for viscous gravity currents from microscopic simulation data, Dynamical Systems–Based Neural Networks, A priori error estimate of deep mixed residual method for elliptic PDEs, Deep Ritz method for elliptical multiple eigenvalue problems, A novel growing wavelet neural network algorithm for solving chemotaxis systems with blow‐up, Feature engineering with regularity structures, Discontinuity computing using physics-informed neural networks, Local randomized neural networks with discontinuous Galerkin methods for diffusive-viscous wave equation, Application of machine learning regression models to inverse eigenvalue problems, Variationally mimetic operator networks, The anisotropic graph neural network model with multiscale and nonlinear characteristic for turbulence simulation, A neural network-based enrichment of reproducing kernel approximation for modeling brittle fracture, Residual-based error corrector operator to enhance accuracy and reliability of neural operator surrogates of nonlinear variational boundary-value problems, Discovering a reaction-diffusion model for Alzheimer's disease by combining PINNs with symbolic regression, Complete flow characterization from snapshot PIV, fast probes and physics-informed neural networks, Multi-level neural networks for accurate solutions of boundary-value problems, A neural network finite element method for contact mechanics, Learning stiff chemical kinetics using extended deep neural operators, A nonlinear-manifold reduced-order model and operator learning for partial differential equations with sharp solution gradients, Some models are useful, but how do we know which ones? Towards a unified Bayesian model taxonomy, Multi-output physics-informed neural network for one- and two-dimensional nonlinear time distributed-order models, Bayesian deep learning for partial differential equation parameter discovery with sparse and noisy data, Physics-constrained data-driven variational method for discrepancy modeling, Feature-adjacent multi-fidelity physics-informed machine learning for partial differential equations, Accelerating hypersonic reentry simulations using deep learning-based hybridization (with guarantees), A hybrid data-driven-physics-constrained Gaussian process regression framework with deep kernel for uncertainty quantification, High Order Deep Domain Decomposition Method for Solving High Frequency Interface Problems, Convergence rates for ansatz‐free data‐driven inference in physically constrained problems, The use of physics-informed neural network approach to image restoration via nonlinear PDE tools, A numerical comparison of simplified Galerkin and machine learning reduced order models for vaginal deformations, Data-driven wave solutions of (2+1)-dimensional nonlinear dispersive long wave equation by deep learning, Optimal control for sampling the transition path process and estimating rates, Exploring two-dimensional internal waves: a new three-coupled Davey-Stewartson system and physics-informed neural networks with weight assignment methods, Gradient-enhanced physics-informed neural networks based on transfer learning for inverse problems of the variable coefficient differential equations, Ensemble forecasts in reproducing kernel Hilbert space family, An Adaptive Physics-Informed Neural Network with Two-Stage Learning Strategy to Solve Partial Differential Equations, Discrepancy Modeling Framework: Learning Missing Physics, Modeling Systematic Residuals, and Disambiguating between Deterministic and Random Effects, Less Emphasis on Hard Regions: Curriculum Learning of PINNs for Singularly Perturbed Convection-Diffusion-Reaction Problems, Gaussian process regression as a surrogate model for the computation of dispersion relations, Spectral operator learning for parametric PDEs without data reliance, Physics-informed graph neural network emulation of soft-tissue mechanics, A deep learning method for multi-material diffusion problems based on physics-informed neural networks, A complete physics-informed neural network-based framework for structural topology optimization, 3D elastic wave propagation with a factorized Fourier neural operator (F-FNO), The random feature method for solving interface problems, A super-real-time three-dimension computing method of digital twins in space nuclear power, Convergence Analysis of the Deep Galerkin Method for Weak Solutions, AONN: An Adjoint-Oriented Neural Network Method for All-At-Once Solutions of Parametric Optimal Control Problems, Neural Control of Parametric Solutions for High-Dimensional Evolution PDEs, Learning the Dynamics for Unknown Hyperbolic Conservation Laws Using Deep Neural Networks, A machine learning approach to enhance the SUPG stabilization method for advection-dominated differential problems, Dynamics of Controllable Matter-Wave Solitons and Soliton Molecules for a Rabi-Coupled Gross–Pitaevskii Equation with Temporally and Spatially Modulated Coefficients, Artificial neural network-augmented stabilized finite element method, Optimal Dirichlet boundary control by Fourier neural operators applied to nonlinear optics, Discovering first principle of behavioural change in disease transmission dynamics by deep learning, A new numerical approach method to solve the Lotka-Volterra predator-prey models with discrete delays, wPINNs: Weak Physics Informed Neural Networks for Approximating Entropy Solutions of Hyperbolic Conservation Laws, A Reduced Order Schwarz Method for Nonlinear Multiscale Elliptic Equations Based on Two-Layer Neural Networks, Differentiable hybrid neural modeling for fluid-structure interaction, A hybrid approach for solving the gravitational \(N\)-body problem with artificial neural networks, Learning physical models that can respect conservation laws, Multi-layer neural networks for data-driven learning of fractional difference equations' stability, periodicity and chaos, Is the neural tangent kernel of PINNs deep learning general partial differential equations always convergent?, Physics-informed ConvNet: learning physical field from a shallow neural network, The mathematics of artificial intelligence, Solving inverse problems with deep learning, Data-driven Whitney forms for structure-preserving control volume analysis, Derivative-informed neural operator: an efficient framework for high-dimensional parametric derivative learning, CNN-DP: composite neural network with differential propagation for impulsive nonlinear dynamics, Operator approximation of the wave equation based on deep learning of Green's function, Physical informed neural networks with soft and hard boundary constraints for solving advection-diffusion equations using Fourier expansions, Piecewise DMD for oscillatory and Turing spatio-temporal dynamics, Error assessment of an adaptive finite elements -- neural networks method for an elliptic parametric PDE, Residual-based attention in physics-informed neural networks, Optimization of physics-informed neural networks for solving the nolinear Schrödinger equation, A new computationally simple approach for implementing neural networks with output hard constraints, Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions, Convergence rate of DeepONets for learning operators arising from advection-diffusion equations, ExSpliNet: An interpretable and expressive spline-based neural network

Uses Software

• L-BFGS
• Chebfun
• TensorFlow
• DiffSharp
• Spearmint
• ImageNet
• Adam
• Omniglot
• AlexNet

Cites Work

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